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Convergence of the Weil-Petersson metric on the Teichmüller space of bordered Riemann surfaces
2015 (English)In: Communications in Contemporary Mathematics, ISSN 0219-1997Article in journal (Refereed) Published
Abstract [en]

Consider a Riemann surface of genus g bordered by n curves homeomorphic to the unit circle, and assume that 2g−2+n > 0. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space.We show that any tangent vector can be represented as the derivative of a holomor-phic curve whose representative Beltrami differentials are simultaneously in L2 and L ,and furthermore that the space of (−1,1) differentials in L2 ∩L decomposes as a directsum of infinitesimally trivial differentials and L2 harmonic (−1,1) differentials. Thus thetangent space of this Teichmüller space is given by L2 harmonic Beltrami differentials.We conclude that this Teichmüller space has a finite Weil-Petersson Hermitian metric.Finally, we show that the aforementioned Teichmüller space is locally modeled on a spaceof L2 harmonic Beltrami differentials.

Place, publisher, year, edition, pages
World Scientific, 2015.
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URN: urn:nbn:se:uu:diva-284916OAI: oai:DiVA.org:uu-284916DiVA: diva2:920966
Available from: 2016-04-19 Created: 2016-04-19 Last updated: 2016-09-28

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Staubach, Wolfgang
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